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The Bungee Jumper: A Comparison of Predicted and Measured Values Hubert Biezeveld, OSG West-Friesland, Hoorn, The Netherlands T he greater-than-g acceleration of a bungee jumper discussed in a previous article in this journal by Kagan and Kott1 led to many lively discussions among Dutch physics teachers. These inspired me to look for an inexpensive experimental setup, suitable for use in a high school physics class, that can be used to confirm that indeed the acceleration is greater than g. In this paper I describe an exercise to compare the predicted and the measured graphs for the displacement y(t) of the jumper and the force Fb(t) exerted by the bungee on the bridge to which it is fastened. In my apparatus, the “bungee” consists of a light chain and the “jumper” is a small piece of brass. Data collection and the calculation of predicted values were carried out using Coach.2 The analysis reliably leads to the conclusion that the acceleration of the falling jumper does indeed exceed g. Theory A real bungee jumper (a person of mass M) is tied to an elastic cord (bungee) of mass m and length L. The other end of the rope is tied to a fixed point — perhaps on a bridge. Starting from rest on the bridge, the jumper accelerates downward to speed v after falling a distance y. Using conservation of mechanical energy, Kagan and Kott show that while the cord is still slack: (4ML + 2mL – my) v2 = gy . (mL – my + 2mL) And differentiating: 238 (1) my(4ML+2mL–my) a = g 1+ . 2(mL–my+2ML)2 冤 冥 (2) A solution for y(t) may be obtained in terms of elliptic integrals.3 This is inappropriate for the introductory-level physics course, however. Numerical integration using, for example, a spreadsheet is certainly preferable. Commenting about the counterintuitive conclusion that a > g, Kagan and Kott write: “Some insight can be gleaned from thinking about the fact that the portion of cord at the bottom of the loop is coming to rest. Since it was moving downward, an upward force is required on this part of the system.” This remark put me on the track to find a simple algorithm for determining numerically the displacement y(t) of the jumper and also the force Fb(t) exerted by the bungee on the bridge. My analysis is limited to the time during which the bungee remains slack. Algorithm The method consists of N successive loops in which we continually calculate new values for t, y, and Fb. We start with the values of the various constants employed in our experimental setup. The length and mass of the chain (bungee) are L = 1.51 m and m = 0.290 kg. The mass of the brass cylinder (jumper) is 0.057 kg. The linear density of the bungee is = m/L. We divide the length L into N (N = 10,000 for the calculations described here) pieces of length dy = L/N, each having mass dy. After the values of the constants have been set, the iterative process may be initiated. DOI: 10.1119/1.1564507 THE PHYSICS TEACHER ◆ Vol. 41, April 2003 Fig. 2. A brass chain is used for the bungee. • Fig. 1(a). Bungee jumper system at time t. Fig. 1(b). The system at time t + dt. • The following is an annotated list of the steps contained in each of the N calculation loops: • • • • • • At the start of a loop, the elapsed time is t, and the jumper has fallen a distance y. Referring to Fig. 1(a), we see that a length ½(L + y) of the bungee (on the left side) is at rest. On the right side we have both the jumper and a length ½(L – y) of bungee moving with velocity v1 [see Eq. (1)]. We add dy to y and write y := y + dy (:= stands for “the old value is replaced by ...”). From this new y we calculate a new velocity v2 using Eq. (1). The mean velocity vm = (v1 + v2)/2 is calculated. The time dt needed for the displacement dy is dt = dy/vm. This value of dt is added to t, and t := t + dt. When the jumper descends the distance dy a portion ½dy of the bungee having mass ½dy and velocity v1 at the right side of the bottom of the loop [Fig. 1(b)] comes to rest on the left side. To bring this mass to rest, a force F = (½dy)v1/dt (Newton’s second law) is required. Since the tension in the bungee is continuous,3 half of this force THE PHYSICS TEACHER ◆ Vol. 41, April 2003 is supplied by the left side and the other half by the right side of the bungee. The total force Fb on the bridge may be written as: Fb = ½(L + y)g + ½ (½dy)v1/dt. The first term is the static weight of the left side of the bungee; the second term is half of the dynamic force just calculated. This force Fb can be measured with a force sensor. To start a new loop we simply replace v1 by v2, i.e., v1 := v2. Measurements and Comparison to Predicted Values As mentioned earlier, in this experiment the jumper is made of brass. It is actually connected to two chains (bungees) (see Fig. 2) in order to reduce the possibility of horizontal motion during the fall. The jumper is constructed so that it can slide down a constantan wire across which a voltage of 5.00 V is applied. This setup constitutes a potentiometer that can be calibrated as a y-sensor. See Appendix for a detailed description. Figure 3 shows a comparison of my measured y(t) for the jumper (red) and y(t) for free fall, i.e., y = 4.9t 2 (green). It is evident that the bungee jumper does indeed have a greater-than-g acceleration. Figure 4 shows the measured y(t) (red) and y(t) as determined using Eq. (1) and the algorithm described (green). The agreement is very good. The force Fb exerted by the fixed support on one of the two chains was measured with a standard force 239 Fig. 3. Experimental y(t) graph for the jumper (red) and for a mass in free fall [y(t) = 4.9t2] (green). Fig. 5. Fb(t) graphs for a bungee jump (red: experimental and green: theoretical). bungee jump. Predicted and measured values agree very well. Acknowledgments I wish to thank my colleagues René Peerdeman and Joost Osterhaus for making the figures and reading the text, and also an anonymous referee for his/her useful and encouraging remarks. Appendix Fig. 4. y(t) graphs for a bungee jump (red: experimental and green: theoretical). sensor.4 Figure 5 shows the predicted Fb(t) graph (green) and the measured Fb(t) graph (red). Again, the curves are seen to be in good agreement. Summary I have described an exercise for determining both theoretically and experimentally the displacement y(t) of a simulated bungee jumper. The apparatus and methods are appropriate for use in an introductorylevel physics course. The modeling in this article is performed using one of the modules of Coach, but other spreadsheets or a BASIC program would also work. Any interface that can handle voltages between 0 and 5 V can be used to feed the data into a computer. The advantage of Coach is that it allows the possibility of displaying the predicted graph and the measured graph on the same axes. With some modifications of the bungee system, the measurements could be made using a motion sensor. I’ve also shown that it is possible to investigate forces that act during a 240 The y-ssensor My y-sensor is cheap and took less than one hour to make. It is, in fact, a potentiometer whose sliding element is the cylindrical brass “jumper” (4.0 cm in length and 1.5 cm in diameter). I drilled an axial hole (6 mm in diameter) into the bottom of the cylinder extending almost to the top and lined the hole with a piece of plastic tubing. I drilled a smaller (1.0 mm diameter) axial hole in the top of the cylinder and passed a constantan wire (0.4 mm diameter) through this hole and the rest of the cylinder. The constantan wire is stretched vertically between two fixed points (see Fig. 6). A stabilized voltage of 5.00 V is applied across the wire and a small bolt is attached to the top of the cylinder. A very thin copper wire is connected to this bolt and lightly wrapped twice around the constantan wire. It’s critical that this wrapping be done carefully — not overly tight (too much friction during the fall) and not too loose (poor electrical contact). To repress possible pulses caused by erratic contact, I connect an 0.5-F capacitor across the input of the computer. As the cylinder falls, the potential difference between the sliding contact and ground can be THE PHYSICS TEACHER ◆ Vol. 41, April 2003 Fig. 6. Schematic diagram of the y-sensor. etcetera... fed into any interface that can handle voltages between 0 and 5 V. I used CMA’s CoachLab 2 in combination with Coach, which gave me the possibility of generating theoretical and experimental graphs in one integrated experiment. It would also be possible to use the Vernier differential voltage probe in combination with LabPro. References 1. D. Kagan and A. Kott, “The greater-than-g acceleration of a bungee jumper,” Phys. Teach. 34, 368–373 (Sept. 1996). 2. Coach is a versatile activity-based environment that makes it possible to: (a) collect data with all kinds of sensors; (b) collect data from video-clips; (c) process these data with advanced mathematical tools; and (d) compare the experimental results with predictions obtained by modeling. Coach also supports Texas Instruments CBLs and Vernier LabPro. Coach is a product of CMA, AMSTEL Institute, Universiteit van Amsterdam. See http://www.cma.science.uva.nl/english, or http://www.harris-educational.com/Probeware. 3. M.G. Calkin and R.H. March, “The dynamics of a falling chain: I,” Am. J. Phys. 57, 154–157 (1989). 4. The CMA Dual-Range Force Sensor B0362bt. The Vernier Dual-Range Force Sensor DFS-BTA could also be used. PACS codes: 02.60, 06.90, 46.20B, 46.05B Hubert Biezeveld has an M.S. degree in physics from the University of Amsterdam. He has been teaching high school physics since 1968 and co-authored a high school physics textbook Scoop (Dutch word for Scope!). He also was the editor of Faraday, the journal for Dutch teachers in chemistry and physics, and received the Minnaert Award for his didactic activities. OSG West-Friesland, Hoorn, The Netherlands; hubert.biezeveld@planet.nl etcetera... Editor Albert A. Bartlett, Department of Physics, University of Colorado, Boulder, CO 80309-0390 Multiple-Choice Question1 Which of the following is not an answer to this question? a) All of the responses below b) This one c) Some of the above d) All of the above e) None of these 1. Donald E. Simanek, The Vector 20, 29 (Fall 1991). 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